A249336 a(1) = 1; for n>1, a(n) = number of values k in range 1 .. n-1 such that {sum of prime indices in the prime factorization of a(k)} = {sum of prime indices in the prime factorization of a(n-1)}, both counted with multiplicity.
1, 1, 2, 1, 3, 1, 4, 2, 2, 3, 3, 4, 5, 1, 5, 2, 4, 6, 3, 7, 1, 6, 4, 8, 5, 6, 7, 2, 5, 8, 9, 3, 9, 4, 10, 5, 10, 6, 11, 1, 7, 7, 8, 12, 9, 10, 11, 2, 6, 13, 1, 8, 14, 3, 11, 4, 12, 12, 13, 2, 7, 14, 5, 15, 6, 16, 15, 7, 16, 17, 1, 9, 18, 8, 17, 2, 8, 18, 9, 19, 1, 10, 20, 10, 21, 3, 13, 4, 14, 11, 12, 22, 5, 19, 2, 9, 23, 1
Offset: 1
Keywords
Examples
a(1) = 1 by definition. For n = 2, we see that a(n-1) = a(1) = 1, the sum of whose prime indices is 0, and the only integer k for which A056239(k) = 0 is 1, and 1 occurs once among the terms a(1) .. a(1), thus a(2) = 1 also. For n = 3, we see that a(n-1) = a(2) = 1 occurs two times among the terms a(1) .. a(2), thus a(3) = 2. For n = 4, we see that a(n-1) = a(3) = 2, and A056239(2) = 1, and so far there are no other terms than a(3) in a(1) .. a(3) which would result the same sum, thus a(4) = 1. For n = 5, we see that a(n-1) = a(4) = 1 occurs three times in a(1) .. a(4), thus a(5) = 3. For n = 6, we see that a(n-1) = a(5) = 3, and A056239(3) = 2 (as 3 = p_2), and so far there are no other terms than a(5) in a(1) .. a(5) which would result the same sum, thus a(6) = 1. For n = 7, we see that a(n-1) = a(6) = 1 occurs four times in a(1) .. a(6), thus a(7) = 4. For n = 8, we see that a(n-1) = a(7) = 4, and A056239(4) = 2 (as 4 = p_1 * p_1), and so far among the terms a(1) .. a(7) only a(5) results in the same sum, thus a(8) = 2.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..12580
Crossrefs
Programs
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PARI
A049084(n) = if(isprime(n), primepi(n), 0); \\ This function from Charles R Greathouse IV A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * A049084(f[i,1]))); } A249336_write_bfile(up_to_n) = { my(counts, n, a_n); counts = vector(up_to_n); a_n = 1; for(n = 1, up_to_n, write("b249336.txt", n, " ", a_n); counts[1+A056239(a_n)]++; a_n = counts[1+A056239(a_n)]); }; A249336_write_bfile(12580);
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Scheme
;; With memoization-macro definec from Antti Karttunen's IntSeq-library: (definec (A249336 n) (if (<= n 1) n (let ((s (A056239 (A249336 (- n 1))))) (let loop ((i (- n 1)) (k 0)) (cond ((zero? i) k) ((= (A056239 (A249336 i)) s) (loop (- i 1) (+ k 1))) (else (loop (- i 1) k))))))) ;; Slow, quadratic time implementation.
Comments